{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,10]],"date-time":"2026-01-10T04:19:25Z","timestamp":1768018765812,"version":"3.49.0"},"reference-count":10,"publisher":"University of Zielona G\u00f3ra, Poland","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2012,6,1]]},"abstract":"<jats:title>Infinite-dimensional Sylvester equations: Basic theory and application to observer design<\/jats:title>\n        <jats:p>This paper develops a mathematical framework for the infinite-dimensional Sylvester equation both in the differential and the algebraic form. It uses the implemented semigroup concept as the main mathematical tool. This concept may be found in the literature on evolution equations occurring in mathematics and physics and is rather unknown in systems and control theories. But it is just systems and control theory where Sylvester equations widely appear, and for this reason we intend to give a mathematically rigorous introduction to the subject which is tailored to researchers and postgraduate students working on systems and control. This goal motivates the assumptions under which the results are developed. As an important example of applications we study the problem of designing an asymptotic state observer for a linear infinite-dimensional control system with a bounded input operator and an unbounded output operator.<\/jats:p>","DOI":"10.2478\/v10006-012-0018-5","type":"journal-article","created":{"date-parts":[[2012,6,29]],"date-time":"2012-06-29T14:49:17Z","timestamp":1340981357000},"page":"245-257","source":"Crossref","is-referenced-by-count":10,"title":["Infinite-dimensional Sylvester equations: Basic theory and application to observer design"],"prefix":"10.61822","volume":"22","author":[{"given":"Zbigniew","family":"Emirsaj\u0142ow","sequence":"first","affiliation":[]}],"member":"37438","reference":[{"issue":"3","key":"1","doi-asserted-by":"crossref","first-page":"371","DOI":"10.1007\/s002330010082","article-title":"On implemented semigroups","volume":"63","author":"J. Alber","year":"2001","journal-title":"Semigroup Forum"},{"issue":"2","key":"2","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1093\/qmath\/45.2.133","article-title":"Spectral properties of the operator equation <i>ax<\/i> + <i>xb<\/i> = <i>y<\/i>","volume":"45","author":"W. Arendt","year":"1994","journal-title":"Quarterly Journal of Mathematics"},{"key":"3","volume-title":"Linear Systems Theory and Design","author":"C.-T. Chen","year":"1984"},{"key":"4","volume-title":"Implemented Semigroup in Analysis of Infinite-Dimensional Sylvester and Lyapunov equations","author":"Z. Emirsaj\u0142ow","year":"2005"},{"issue":"1","key":"5","first-page":"1","article-title":"From PDEs with a boundary control to the abstract state equation with an unbounded input operator","volume":"7","author":"Z. Emirsaj\u0142ow","year":"2000","journal-title":"European Journal of Control"},{"issue":"4","key":"6","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1080\/00207170500093960","article-title":"On application of the implemented semigroup to a problem arising in optimal control","volume":"78","author":"Z. Emirsaj\u0142ow","year":"2005","journal-title":"International Journal of Control"},{"key":"7","volume-title":"One-Parameter Semigroups for Linear Evolution Equations","author":"J. Engel","year":"2000"},{"key":"8","volume-title":"Lyapunov Matrix Equation in Systems Stability and Control","author":"Z. Gajic","year":"2000"},{"key":"9","unstructured":"Kuehnemund, F. (2001). <i>Bi-continuous Semigroups on Spaces with Two Topologies: Theory and Applications<\/i>, Ph.D. thesis, Eberhard-Karls-Univerit\u00e4t Tuebingen, Tuebingen."},{"issue":"7","key":"10","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1007\/BF02571546","article-title":"The operator equation <i>ax<\/i> - <i>xb<\/i> = <i>c<\/i> with unbounded operators <i>a<\/i> and <i>b<\/i> and related Cauchy problems","volume":"208","author":"V. Phong","year":"1991","journal-title":"Mathematische Zeitschrift"}],"container-title":["International Journal of Applied Mathematics and Computer Science"],"original-title":[],"link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/amcs\/22\/2\/article-p245.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/view\/j\/amcs.2012.22.issue-2\/v10006-012-0018-5\/v10006-012-0018-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,2,29]],"date-time":"2024-02-29T10:27:29Z","timestamp":1709202449000},"score":1,"resource":{"primary":{"URL":"https:\/\/content.sciendo.com\/doi\/10.2478\/v10006-012-0018-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,6,1]]},"references-count":10,"journal-issue":{"issue":"2"},"URL":"https:\/\/doi.org\/10.2478\/v10006-012-0018-5","relation":{},"ISSN":["1641-876X"],"issn-type":[{"value":"1641-876X","type":"print"}],"subject":[],"published":{"date-parts":[[2012,6,1]]}}}